Sobolev homeomorphisms and composition operators on homogeneous Lie groups

In this article, we study Sobolev homeomorphisms and composition operators on homogeneous Lie groups. We prove that a measurable homeomorphism $\phi :\mathrm{\Omega }\to \tilde{\mathrm{\Omega }}$ belongs to the Sobolev space $L_q^1(\mathrm{\Omega };\tilde{\mathrm{\Omega }})$, $1\le q<\mathrm{\infty }$, if and only if $\phi$ generates a bounded composition operator on Sobolev spaces.